tgt

## Saturday, 9 August 2014

### chapter 1 concept of limits

The concept of limits forms the basis of calculus and is a very powerful one. Both differential and integral calculus are based on this concept and as such, limits need to be studied in good detail.
This section contains a general, intuitive introduction to limits.
Consider a circle of radius $r$.
We know that the area of this circle is $\pi {r^2}$ . How$?$
The ancient Greeks derived this result using the concept of limits.
To see how, recall the definition of $\pi$.
 $\begin{array}{l} \pi = \dfrac{{{\rm{length}}\,{\rm{of}}\,{\rm{circumference}}}}\,{{{\rm{length}}\,{\rm{of}}\,{\rm{diameter}}}}\,\\ \pi = \dfrac{c}{d} = \dfrac{c}{{2r}}\\ c = 2\pi r \end{array}$
With this definition in hand, the Greeks divided the circle as follows (like cutting a cake or a pie):
Now they took the different pieces of this ‘pie’ and placed them as follows:
See what happens if the number of cuts are increased
The figure on the right side starts resembling a rectangle as we increase the number of cuts to the circle. The sequence of curves that joins $x$to $y$ starts becoming more and more of a straight line with the same total length $\pi r$ .
What happens as we increase the number of cuts indefinitely, or equivalently, we decrease $\theta$ indefinitely? The figure ‘almost’ becomes a rectangle, though never becoming a rectangle exactly. The area ‘almost’ becomes $\pi r \times r = \pi {r^2}.$
In the language of limits, we say that the figure tends to a rectangle or the area $A$ tends to $\pi {r^2}$ or the limiting value of area is $\pi {r^2}.$
In standard terminology.
 $\mathop {{\rm{lim}}}\limits_{\theta \to 0} {\rm{A}} = \pi {r^2}$